For example, Paul said the Tea Party is largely responsible for the debate about the nation's $13.7 trillion debt.

Five days after delivering a 12-point pounding to Democrat Jack Conway in Kentucky, Paul avoided offering any new specifics about how he would balance the federal budget but said he would like to have a seat on the Senate Budget Committee.

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"My hope is to be on the Budget Committee and to go through all of these numbers and, by January, to have a balanced budget that I will introduce," Paul said. "I want there to be a Republican alternative. Whether it wins or not, I want the Republican message to be one of balanced budgets."

Meanwhile, Senate Minority Leader Mitch McConnell — Kentucky's senior senator — was on CBS talking about extending tax cuts for those who make $250,000 or more a year, a move that would amount to $700 billion over the next 10 years that the government would not collect.

McConnell said on Face the Nation he is willing to work with President Barack Obama to get a tax-cut extension "of some kind" for those making $250,000 or more a year.

"Let me make sure everybody understands what we're talking about here. These aren't tax cuts; these are tax increases. Tax increases in the middle of a recession," McConnell said.

Asked by host Bob Schieffer how he would pay for the tax cuts, McConnell borrowed the answer Paul used for the same question during his campaign.

"Bob, it only costs $700 billion if you consider it the government's money," McConnell replied. "This is our money. This has been the tax rate for almost a decade."

Both men also spoke about budget earmarks. Paul reiterated his support for ending earmarks and his pledge not to use them to funnel money back to Kentucky projects.

McConnell said he would be "willing to consider" a proposal that bans all lawmakers from putting earmarks in the budget, but called the issue "exasperating."

"The problem is it doesn't save any money," McConnell said. "It's an argument about discretion. What we really need to do, Bob, is to concentrate on reducing spending and reducing debt. And this debate doesn't save any money, which is why it's kind of exasperating to some of us who really want to cut spending and get the federal government's discretionary accounts under control."

On the issue of repealing the health care law, McConnell and Paul had the same message.

"This was a huge, huge issue in the election last Tuesday," McConnell said. "A vast majority of Americans feel very, very uncomfortable with this new bill. People who supported us, political independents, want it repealed and replaced with something else. I think we owe it to them to try."

Even if Republicans were able to sway enough Democrats to support a full repeal, it would face a certain veto from Obama.

"Admittedly, it will be difficult with him in the White House," McConnell said. "But if we can put a full repeal on his desk and replace it with the kind of common-sense forms that we were advocating during the debate to reduce spending, we owe it to the American people to do that."

Paul also vowed to vote against lifting the federal government's debt ceiling.

"I think that we need to send a message — we need to send a strong message ..." Paul said.

"The government would default, then," Amanpour responded.

"Well, only if we won the vote, would they default," Paul said.

"So you think it won't pass?" Amanpour asked.

"You know, I think it's unlikely," Paul said. "There are people who vote against the debt ceiling every time to send a message that adding more debt is wrong. I think we shouldn't add more debt. I think we should immediately start cutting spending."

He went on to outline a handful of general spending cuts that he often spoke about on the campaign trail, including a federal hiring freeze.